![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatlej2 | Structured version Visualization version GIF version |
Description: A join's second argument is less than or equal to the join. Special case of latlej2 17262 to show an atom is on a line. (Contributed by NM, 15-May-2013.) |
Ref | Expression |
---|---|
hlatlej.l | ⊢ ≤ = (le‘𝐾) |
hlatlej.j | ⊢ ∨ = (join‘𝐾) |
hlatlej.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
hlatlej2 | ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlatlej.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | hlatlej.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | hlatlej.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 1, 2, 3 | hlatlej1 35164 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑃)) |
5 | 4 | 3com23 1121 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑃)) |
6 | 2, 3 | hlatjcom 35157 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
7 | 5, 6 | breqtrrd 4832 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 class class class wbr 4804 ‘cfv 6049 (class class class)co 6813 lecple 16150 joincjn 17145 Atomscatm 35053 HLchlt 35140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-lub 17175 df-join 17177 df-lat 17247 df-ats 35057 df-atl 35088 df-cvlat 35112 df-hlat 35141 |
This theorem is referenced by: 2llnne2N 35197 cvrat3 35231 cvrat4 35232 hlatexch3N 35269 hlatexch4 35270 dalem3 35453 dalem25 35487 lnatexN 35568 lncmp 35572 2llnma3r 35577 paddasslem5 35613 dalawlem3 35662 dalawlem6 35665 dalawlem7 35666 dalawlem12 35671 lhp2atne 35823 lhp2at0ne 35825 4atexlemunv 35855 cdlemc2 35982 cdlemc5 35985 cdleme3h 36025 cdleme7 36039 cdleme9 36043 cdleme11c 36051 cdleme11dN 36052 cdleme11j 36057 cdleme16b 36069 cdleme17b 36077 cdleme18a 36081 cdleme18b 36082 cdleme18c 36083 cdleme19a 36093 cdleme20d 36102 cdleme20j 36108 cdleme21ct 36119 cdleme22a 36130 cdleme22e 36134 cdleme22eALTN 36135 cdleme35b 36240 cdlemg9a 36422 cdlemg12a 36433 cdlemg13a 36441 cdlemg17a 36451 cdlemg17g 36457 cdlemg18c 36470 cdlemg33b0 36491 cdlemg46 36525 cdlemh1 36605 cdlemh 36607 cdlemk4 36624 cdlemki 36631 cdlemksv2 36637 cdlemk12 36640 cdlemk15 36645 cdlemk12u 36662 cdlemkid1 36712 dia2dimlem1 36855 dia2dimlem3 36857 cdlemn10 36997 dihjatcclem1 37209 |
Copyright terms: Public domain | W3C validator |